A gear pump could be said to include a first gear and a second gear intermeshed with the first gear. An inlet side is configured to have an inlet connection connected thereto. A discharge side is configured to have a first discharge connection connected thereto. At least one shaft is in operable communication with each of the first and second gears. A bearing is configured to support at least one of the shafts via an inner bore and having an outer peripheral surface. A valve bore is formed into bearing between the inner bore and the outer peripheral surface. A second discharge connection is formed into the bearing. A tap provides fluidic communication between the valve bore and a pad defined in the inner bore. A valve is positioned in the valve bore. The valve includes a moving valve member. A spring biases the moving valve member in a direction toward the second discharge connection. A fuel pump system is also disclosed.